Тема №1: Дискретные структуры, примеры

The 1st theme: Discrete structures, examples

Teacher:

Dotsent, PhD Sabirov Karimjon Kabildjanovich

Introduction

Mathematics

• abstractly describes the material world
• does not come off the real world
• is applied to theoretical physics, quantum mechanics, information technologies
• development led to the emergence and improvement of new directions

Discrete structures

• Discrete structures are the study of processes occurring in nature or in some objects based on their state at some discrete moments of time or their discrete points

Example: the set of natural numbers

the number 3 secede with neighbors 2 and 4

Example for non-discrete is the interval

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Mathematical structures

Mathematical structures can be classified into three main types:

• algebraic structures
• structures of orders
• topological structures

Algebraic structures

If a rule is given for any two elements of the set, for uniquely determining the third element, then this is called the composition law.

A structure defined on the basis of the composition law is called an algebraic structure.

Axiomatic introduction of the operation:

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Examples on algebraic structures

• The set of natural numbers relative to the operations of addition and multiplication
• The set of vectors in the plane relative to the addition operation
• The set of matrices relative to the addition operation
• Boolean structures (logical operation structures)

Structures of orders

• If a relation 𝜌 is given between elements of some set X as follows

then it is said that the order structures are defined in the set X.

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Examples on order structures

• Relation ≤ on the set of real numbers
• A relation ⊂ that defines a subset of sets

Topological structures

If on the set X the operation "[]" is given in the following way, then the topological structures are called:

• If Y is the only element, [Y]=Y

If [A] = A, then A is a closed set

[X] - X is the closure of the set

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Thank you for attentions!